Pith. sign in

REVIEW 2 minor 68 references

Another look at a notion of fractional mass in codimension two

T0 review · 0 major / 2 minor · reviewed 2026-07-02 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The fractional s-mass for codimension-two currents defined via energy minimization with Jacobian constraint agrees with the weak linking definition and satisfies equi-coercivity plus Gamma-convergence.

desk verdict The paper shows the two s-mass definitions agree and extracts a Minkowski dimension bound on singularities from the Gamma-limit. read the letter →

arxiv 2607.00810 v1 pith:AQA57524 submitted 2026-07-01 math.DG math.AP

Michele Caselli , Mattia Freguglia , Nicola Picenni This is my paper · ORCID
classification math.DGmath.AP
keywords fractionalmasscodimension-twocurrentsJacobianconstraintGamma-convergenceflattopologys-harmonicmapsregularityRiemannianmanifolds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper examines a fractional s-mass for codimension-two currents on closed Riemannian manifolds. This mass arises from minimizing an energy functional subject to a prescribed Jacobian constraint. The authors prove that the resulting functional is equi-coercive and Gamma-converges with respect to the flat topology on general such currents. They also establish that this definition produces the same values as an earlier definition based on weak linking, showing the mass is insensitive to the choice of singularity prescription. For any fixed s they obtain improved regularity for minimizing s-harmonic maps with vanishing Jacobian whose singular sets have Minkowski dimension at most n-3.

What carries the argument

The s-mass obtained by minimizing energy under a prescribed Jacobian constraint on codimension-two currents

What would settle it

A codimension-two current on a closed Riemannian manifold for which the value of the energy-minimizing s-mass under the Jacobian constraint differs from the value obtained via the weak linking definition.

Watch

Extended reading notes

Core claim

The fractional s-mass defined via energy minimization with a prescribed Jacobian constraint on general codimension-two currents agrees with the s-mass defined via weak linking. This mass is equi-coercive and Gamma-converges with respect to the flat topology. For fixed s, s-harmonic maps that minimize among maps with vanishing Jacobian have improved regularity and their singular set has Minkowski dimension at most n-3.

Load-bearing premise

That the energy minimization problem with prescribed Jacobian constraint yields a well-defined mass for general codimension-two currents.

Editorial extensions

If this is right

  • The value of the s-mass does not depend on the method used to prescribe singularities.
  • Bounded s-mass implies compactness of sequences of currents in the flat topology.
  • Gamma-convergence permits passage to the limit inside minimization problems that use the s-mass.
  • Minimizing s-harmonic maps with zero Jacobian have singular sets whose Minkowski dimension is at most n-3.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Variational problems involving currents with prescribed singularities can employ either formulation of the s-mass interchangeably.
  • The dimension bound on singular sets supplies quantitative control that may apply to related minimization problems in codimension two.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript defines a fractional s-mass for codimension-two currents on closed Riemannian manifolds via energy minimization subject to a prescribed Jacobian constraint. It establishes equi-coercivity and Γ-convergence of this s-mass with respect to flat convergence on general codimension-two currents. For fixed s it proves improved regularity for minimizing s-harmonic maps with vanishing Jacobian, showing that the singular set has Minkowski dimension at most n-3. It further shows that this prescribed-Jacobian formulation coincides with the authors' earlier weak-linking definition of s-mass.

Significance. If the proofs hold, the work supplies an independent variational characterization of the s-mass and demonstrates its independence from the choice of singularity prescription. The Γ-convergence and equi-coercivity results furnish a solid variational framework for codimension-two problems in geometric measure theory. The regularity statement strengthens control on the singular set of minimizing maps. Explicit credit is due for the equivalence proof between the two definitions and for deriving the dimension bound from a standard blow-up argument once the Γ-limit is identified.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'several additional results for fixed s' is vague; a brief enumeration of those results would improve readability without lengthening the abstract appreciably.
  2. [Introduction] The manuscript refers to 'the manifold' without an explicit global assumption (closed, orientable, etc.); a single sentence in the introduction clarifying the standing hypotheses on the ambient manifold would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the fractional s-mass for codimension-two currents, including the equi-coercivity, Γ-convergence results, the regularity bound on the singular set, and the equivalence between the prescribed-Jacobian and weak-linking formulations. We note the recommendation for minor revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; equivalence proved independently

full rationale

The paper defines the s-mass via energy minimization subject to a prescribed Jacobian constraint on codimension-two currents, then establishes equi-coercivity and Γ-convergence in the flat topology. It separately proves that this coincides with the authors' prior weak-linking definition by verifying that linking-based minimizers satisfy the Jacobian constraint and matching energy bounds. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-referential definition, or unverified self-citation; the equivalence is derived as an independent verification rather than assumed. The regularity result for singular sets follows from standard blow-up arguments once the Γ-limit is identified. The derivation chain is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Abstract-only; the central claims rest on standard background in geometric measure theory (currents, flat topology, Gamma-convergence) and the specific definition via energy minimization with Jacobian constraint. No free parameters or invented entities are visible.

assumptions (2)
  • domain assumption The ambient space is a closed Riemannian manifold.
    Stated explicitly as the setting for the currents.
  • domain assumption The flat topology is the appropriate topology for studying convergence of codimension-two currents.
    Invoked for the equi-coercivity and Gamma-convergence statements.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Another look at a notion of fractional mass in codimension two." pith.science (2026). https://pith.science/paper/AQA57524

@misc{pith2026260700810,
  author       = {Pith},
  title        = {Pith review of: Another look at a notion of fractional mass in codimension two},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AQA57524}},
  note         = {Machine review of arXiv:2607.00810}
}
abstract

We study a notion of fractional $s$-mass for codimension-two currents on closed Riemannian manifolds, defined via energy minimization with a prescribed Jacobian constraint. We prove equi-coercivity and $\Gamma$-convergence, with respect to the flat topology, of the $s$-mass on general codimension-two currents. We also prove several additional results for fixed $s$. We establish improved regularity for $s$-harmonic maps that are minimizing among competitors with vanishing Jacobian and show that their singular set has Minkowski dimension at most $n-3$. Moreover, we show that the $s$-mass defined via weak linking, as recently introduced by the authors, agrees with the prescribed Jacobian formulation used here, clarifying the extent to which the $s$-mass depends, or ultimately does not depend, on the way singularities are prescribed.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages

  1. [1]

    Alberti, S

    G. Alberti, S. Baldo, and G. Orlandi. Functions with prescribed singularities.J. Eur. Math. Soc. (JEMS), 5(3):275–311, 2003

    work page 2003

  2. [2]

    Alberti, S

    G. Alberti, S. Baldo, and G. Orlandi. Variational convergence for functionals of Ginzburg- Landau type.Indiana Univ. Math. J., 54(5):1411–1472, 2005

    work page 2005

  3. [3]

    Alicandro, N

    R. Alicandro, N. Ansini, A. Braides, A. Piatnitski, and A. Tribuzio.A variational theory of convolution-type functionals. SpringerBriefs on PDEs and Data Science. Springer, Singapore,

    work page

  4. [4]

    Alicandro, A

    R. Alicandro, A. Braides, M. Solci, and G. Stefani. Topological singularities arising from fractional-gradient energies.Math. Ann., 393(1):71–111, 2025

    work page 2025

  5. [5]

    Alicandro and M

    R. Alicandro and M. Cicalese. Variational analysis of the asymptotics of theXYmodel. Arch. Ration. Mech. Anal., 192(3):501–536, 2009

    work page 2009

  6. [6]

    Almgren, W

    F. Almgren, W. Browder, G. Caldini, and C. De Lellis. Optimal smooth approximation of integral cycles.arXiv:2411.17678, 2024

    work page arXiv 2024

  7. [7]

    Ambrosio, G

    L. Ambrosio, G. De Philippis, and L. Martinazzi. Gamma-convergence of nonlocal perime- ter functionals.Manuscripta Math., 134(3-4):377–403, 2011

    work page 2011

  8. [8]

    Antonucci, M

    C. Antonucci, M. Gobbino, M. Migliorini, and N. Picenni. Optimal constants for a nonlocal approximation of Sobolev norms and total variation.Anal. PDE, 13(2):595–625, 2020

    work page 2020

Show all 68 references
  1. [9]

    Arroyo-Rabasa, P

    A. Arroyo-Rabasa, P . Bonicatto, and G. Del Nin. Representation of the total variation as a Γ-limit of BMO-type seminorms.Indiana Univ. Math. J., 73(1):341–365, 2024

    2024

  2. [10]

    M. Badran. Harmonic maps to the circle with higher dimensional singular set.Proc. Lond. Math. Soc. (3), 132(3):Paper No. e70135, 2026

    2026

  3. [11]

    Biccari, M

    U. Biccari, M. Warma, and E. Zuazua. Local elliptic regularity for the Dirichlet fractional Laplacian.Adv. Nonlinear Stud., 17(2):387–409, 2017

    2017

  4. [12]

    Bourgain, H

    J. Bourgain, H. Brezis, and P . Mironescu. Another look at Sobolev spaces. InOptimal control and partial differential equations, pages 439–455. IOS, Amsterdam, 2001. 49

    2001

  5. [13]

    Bourgain, H

    J. Bourgain, H. Brezis, and P . Mironescu.H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation.Publ. Math. Inst. Hautes ´Etudes Sci., (99):1–115, 2004

    2004

  6. [14]

    Bousquet

    P . Bousquet. Topological singularities inW s,p(SN , S1).J. Anal. Math., 102:311–346, 2007

    2007

  7. [15]

    Bousquet and P

    P . Bousquet and P . Mironescu. Prescribing the Jacobian in critical spaces.J. Anal. Math., 122:317–373, 2014

    2014

  8. [16]

    Brezis and P

    H. Brezis and P . Mironescu.Sobolev maps to the circle—from the perspective of analysis, geom- etry, and topology, volume 96 ofProgress in Nonlinear Differential Equations and their Applica- tions. Birkh ¨auser/Springer, New York, 2021

    2021

  9. [17]

    Brezis and H.-M

    H. Brezis and H.-M. Nguyen. The Jacobian determinant revisited.Invent. Math., 185(1):17– 54, 2011

    2011

  10. [18]

    Brezis, A

    H. Brezis, A. Seeger, J. Van Schaftingen, and P .-L. Yung. Families of functionals represent- ing Sobolev norms.Anal. PDE, 17(3):943–979, 2024

    2024

  11. [19]

    Caffarelli, J.-M

    L. Caffarelli, J.-M. Roquejoffre, and O. Savin. Nonlocal minimal surfaces.Comm. Pure Appl. Math., 63(9):1111–1144, 2010

    2010

  12. [20]

    Canevari, V

    G. Canevari, V . P . C. Le, R. Oliver-Bonafoux, and G. Orlandi.Γ-convergence of thep- Dirichlet energy for manifold-valued maps.arXiv:2505.21257, 2025

    2025

  13. [21]

    Caselli, E

    M. Caselli, E. Florit-Simon, and J. Serra. Fractional Sobolev spaces on Riemannian mani- folds.Math. Ann., 390(4):6249–6314, 2024

    2024

  14. [22]

    Caselli, M

    M. Caselli, M. Freguglia, and N. Picenni. Coercivity and Gamma-convergence of thep- energy of sphere-valued Sobolev maps.Comm. Partial Differential Equations, to appear

  15. [23]

    Caselli, M

    M. Caselli, M. Freguglia, and N. Picenni. A nonlocal approximation of the area in codi- mension two.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, to appear

  16. [24]

    Caselli and L

    M. Caselli and L. Gennaioli. Asymptotics ass→0 + of the fractional perimeter on Rie- mannian manifolds.Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear

  17. [25]

    Cesaroni, L

    A. Cesaroni, L. De Luca, M. Novaga, and M. Ponsiglione. Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows.Comm. Partial Differential Equations, 46(7):1344–1371, 2021

    2021

  18. [26]

    H. Chan, S. Dipierro, J. Serra, and E. Valdinoci. Nonlocal approximation of minimal sur- faces: optimal estimates from stability.arXiv:2308.06328, 2023

    2023

  19. [27]

    H. Chan, M. Freguglia, and M. Inversi.Γ-limsup estimate for a nonlocal approximation of the Willmore functional.Calc. Var. Partial Differential Equations, 64(6):Paper No. 181, 44, 2025

    2025

  20. [28]

    Cheeger and A

    J. Cheeger and A. Naber. Quantitative stratification and the regularity of harmonic maps and minimal currents.Comm. Pure Appl. Math., 66(6):965–990, 2013

    2013

  21. [29]

    Cicalese, T

    M. Cicalese, T. Heilmann, A. Kubin, F. Onoue, and M. Ponsiglione. A notion ofs-fractional mass for 1-currents in higher codimension.Math. Ann., 393(2):2157–2186, 2025

    2025

  22. [30]

    D ´avila

    J. D ´avila. On an open question about functions of bounded variation.Calc. Var. Partial Differential Equations, 15(4):519–527, 2002

    2002

  23. [31]

    Federer and W

    H. Federer and W. H. Fleming. Normal and integral currents.Ann. of Math. (2), 72:458–520, 1960

    1960

  24. [32]

    Florit-Simon

    E. Florit-Simon. Weyl law and convergence in the classical limit for min-max nonlocal minimal surfaces.arXiv:2406.12162, 2024

    2024

  25. [33]

    G. Foghem. Robust interpolation inequalities via chebyshev-type integral inequalities, 2026. 50

    2026

  26. [34]

    Gennaioli and G

    L. Gennaioli and G. Stefani. Sharp conditions for the BBM formula and asymptotics of heat content-type energies.Arch. Ration. Mech. Anal., 250(1):Paper No. 8, 46, 2026

    2026

  27. [35]

    Gobbino and N

    M. Gobbino and N. Picenni. Gamma-liminf estimate for a class of non-local approxima- tions of Sobolev and BV norms.J. Funct. Anal., 289(9):Paper No. 111106, 25, 2025

    2025

  28. [36]

    Hajł asz and O

    P . Hajł asz and O. Martio. Traces of Sobolev functions on fractal type sets and characteri- zation of extension domains.J. Funct. Anal., 143(1):221–246, 1997

    1997

  29. [37]

    Han and Y

    Z.-C. Han and Y. Y. Li. Degenerate elliptic systems and applications to Ginzburg-Landau type equations. I.Calc. Var. Partial Differential Equations, 4(2):171–202, 1996

    1996

  30. [38]

    Hardt and F.-H

    R. Hardt and F.-H. Lin. Mappings minimizing theL p norm of the gradient.Comm. Pure Appl. Math., 40(5):555–588, 1987

    1987

  31. [39]

    He, C.-L

    Y. He, C.-L. Xiang, and G.-F. Zheng. A global regularity theory for sphere-valued fractional harmonic maps.Math. Z., 311(4):Paper No. 89, 17, 2025

    2025

  32. [40]

    M.-C. Hong. Asymptotic behavior for minimizers of a Ginzburg-Landau-type func- tional in higher dimensions associated withn-harmonic maps.Adv. Differential Equations, 1(4):611–634, 1996

    1996

  33. [41]

    R. L. Jerrard. Lower bounds for generalized Ginzburg-Landau functionals.SIAM J. Math. Anal., 30(4):721–746, 1999

    1999

  34. [42]

    R. C. Kirby.The topology of4-manifolds, volume 1374 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1989

    1989

  35. [43]

    P . Lahti. A sharp lower bound for a class of non-local approximations of the total variation. Math. Ann., 392(1):469–486, 2025

    2025

  36. [44]

    D. Li. On Kato-Ponce and fractional Leibniz.Rev. Mat. Iberoam., 35(1):23–100, 2019

    2019

  37. [45]

    Lohi and L

    J. Lohi and L. Kettunen. Whitney forms and their extensions.J. Comput. Appl. Math., 393:Paper No. 113520, 19, 2021

    2021

  38. [46]

    Loss and C

    M. Loss and C. Sloane. Hardy inequalities for fractional integrals on general domains.J. Funct. Anal., 259(6):1369–1379, 2010

    2010

  39. [47]

    Maz’ya and T

    V . Maz’ya and T. Shaposhnikova. On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms.J. Math. Pures Appl. (9), 81(9):877–884, 2002

    2002

  40. [48]

    B. Merlet. Two remarks on liftings of maps with values intoS 1.C. R. Math. Acad. Sci. Paris, 343(7):467–472, 2006

    2006

  41. [49]

    Mihaila and B

    C. Mihaila and B. Seguin. A definition of fractionalk-dimensional measure: bridging the gap between fractional length and fractional area.Fract. Calc. Appl. Anal., 28(1):276–306, 2025

    2025

  42. [50]

    Mihaila and B

    C. Mihaila and B. Seguin. First variation of the fractionalk-dimensional measure: extend- ing the concept of nonlocal curvature to submanifolds.arXiv:2606.24043, 2026

    2026 arXiv

  43. [51]

    Milgrom and I

    P . Milgrom and I. Segal. Envelope theorems for arbitrary choice sets.Econometrica, 70(2):583–601, 2002

    2002

  44. [52]

    Millot, M

    V . Millot, M. Pegon, and A. Schikorra. Partial regularity for fractional harmonic maps into spheres.Arch. Ration. Mech. Anal., 242(2):747–825, 2021

    2021

  45. [53]

    Millot and Y

    V . Millot and Y. Sire. On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres.Arch. Ration. Mech. Anal., 215(1):125–210, 2015

    2015

  46. [54]

    D. Mucci. Strong density results for manifold valued fractional Sobolev maps.Ann. Fac. Sci. Toulouse Math. (6), 33(3):581–610, 2024

    2024

  47. [55]

    Nguyen.Γ-convergence, Sobolev norms, and BV functions.Duke Math

    H.-M. Nguyen.Γ-convergence, Sobolev norms, and BV functions.Duke Math. J., 157(3):495–533, 2011. 51

    2011

  48. [56]

    H.-M. Nguyen. Characterizations of the Sobolev norms and the total variation via nonlocal functionals, and related problems.C. R. Math. Acad. Sci. Paris, 363:1429–1455, 2025

    2025

  49. [57]

    Paroni, P

    R. Paroni, P . Podio-Guidugli, and B. Seguin. On the nonlocal curvatures of surfaces with or without boundary.Commun. Pure Appl. Anal., 17(2):709–727, 2018

    2018

  50. [58]

    A. C. Ponce. A new approach to Sobolev spaces and connections toΓ-convergence.Calc. Var. Partial Differential Equations, 19(3):229–255, 2004

    2004

  51. [59]

    V . S. Rychkov. On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains.J. London Math. Soc. (2), 60(1):237–257, 1999

    1999

  52. [60]

    Savin and E

    O. Savin and E. Valdinoci.Γ-convergence for nonlocal phase transitions.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 29(4):479–500, 2012

    2012

  53. [61]

    Savin and E

    O. Savin and E. Valdinoci. Regularity of nonlocal minimal cones in dimension 2.Calc. Var. Partial Differential Equations, 48(1-2):33–39, 2013

    2013

  54. [62]

    B. Seguin. A fractional notion of length and an associated nonlocal curvature.J. Geom. Anal., 30(1):161–181, 2020

    2020

  55. [63]

    J. Serra. Nonlocal minimal surfaces: recent developments, applications, and future direc- tions.SeMA J., 81(2):165–191, 2024

    2024

  56. [64]

    Simon.Lectures on geometric measure theory, volume 3 ofProceedings of the Centre for Math- ematical Analysis, Australian National University

    L. Simon.Lectures on geometric measure theory, volume 3 ofProceedings of the Centre for Math- ematical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983

    1983

  57. [65]

    M. Solci. Nonlocal-Interaction Vortices.SIAM J. Math. Anal., 56(3):3430–3451, 2024

    2024

  58. [66]

    Stern.Variational Theory and Asymptotic Analysis for the Ginzburg-Landau Equations and p-Harmonic Maps

    D. Stern.Variational Theory and Asymptotic Analysis for the Ginzburg-Landau Equations and p-Harmonic Maps. PhD thesis, Princeton University, 2019

    2019

  59. [67]

    Wang, C.-L

    Y.-Y. Wang, C.-L. Xiang, and G.-F. Zheng. Quantitative regularity for minimizing intrinsic fractional harmonic maps.J. Differential Equations, 463:Paper No. 114243, 31, 2026

    2026

  60. [68]

    Whitney.Geometric integration theory

    H. Whitney.Geometric integration theory. Princeton University Press, Princeton, NJ, 1957. MICHELECASELLI University of Sydney – Quadrangle A14, Camperdown, NSW 2006, Australia Princeton University – 304 Washington Rd, Princeton NJ 08540, US E-mail address:mc3147@princeton.edu ...

    1957

Pith tools

Reviewed July 2, 2026 · model on record in the stance chip above.